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Exponential Functions and ModelsPowerPoint Presentation

Exponential Functions and Models

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Exponential Functions and Models. Lesson 5.3. Contrast. Definition. An exponential function Note the variable is in the exponent The base is a C is the coefficient, also considered the initial value (when x = 0). Explore Exponentials.

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### Exponential Functions and Models

Lesson 5.3

Definition

- An exponential function
- Note the variable is in the exponent
- The base is a
- C is the coefficient, also considered the initial value (when x = 0)

Explore Exponentials

- Given f(1) = 3, for each unit increase in x, the output is multiplied by 1.5
- Determine the exponential function

Explore Exponentials

- Graph these exponentials
- What do you think the coefficient C and the base a do to the appearance of the graphs?

Contrast Linear vs. Exponential

- Suppose you have a choice of two different jobs at graduation
- Start at $30,000 with a 6% per year increase
- Start at $40,000 with $1200 per year raise

- Which should you choose?
- One is linear growth
- One is exponential growth

Which Job?

- How do we get each nextvalue for Option A?
- When is Option A better?
- When is Option B better?
- Rate of increase a constant $1200
- Rate of increase changing
- Percent of increase is a constant
- Ratio of successive years is 1.06

Example

- Consider a savings account with compounded yearly income
- You have $100 in the account
- You receive 5% annual interest

View completed table

Compounded Interest

- Completed table

Compounded Interest

- Table of results from calculator
- Set Y= screen y1(x)=100*1.05^x
- Choose Table (♦Y)

- Graph of results

Assignment A

- Lesson 5.3A
- Page 415
- Exercises 1 – 57 EOO

Compound Interest

- Consider an amount A0 of money deposited in an account
- Pays annual rate of interest r percent
- Compounded m times per year
- Stays in the account n years

- Then the resulting balance An

Exponential Modeling

- Population growth often modeled by exponential function
- Half life of radioactive materials modeled by exponential function

Growth Factor

- Recall formulanew balance = old balance + 0.05 * old balance
- Another way of writing the formulanew balance = 1.05 * old balance
- Why equivalent?
- Growth factor: 1 + interest rate as a fraction

Decreasing Exponentials

- Consider a medication
- Patient takes 100 mg
- Once it is taken, body filters medication out over period of time
- Suppose it removes 15% of what is present in the blood stream every hour

Fill in the rest of the table

What is the growth factor?

Decreasing Exponentials

- Completed chart
- Graph

Growth Factor = 0.85

Note: when growth factor < 1, exponential is a decreasing function

Solving Exponential Equations Graphically

- For our medication example when does the amount of medication amount to less than 5 mg
- Graph the functionfor 0 < t < 25
- Use the graph todetermine when

General Formula

- All exponential functions have the general format:
- Where
- A = initial value
- B = growth rate
- t = number of time periods

Typical Exponential Graphs

- When B > 1
- When B < 1

Using e As the Base

- We have used y = A * Bt
- Consider letting B = ek
- Then by substitution y = A * (ek)t
- Recall B = (1 + r) (the growth factor)
- It turns out that

Continuous Growth

- The constant k is called the continuous percent growth rate
- For Q = a bt
- k can be found by solving ek = b

- Then Q = a ek*t
- For positive a
- if k > 0 then Q is an increasing function
- if k < 0 then Q is a decreasing function

Continuous Growth

- For Q = a ek*t Assume a > 0
- k > 0
- k < 0

Continuous Growth

- For the functionwhat is thecontinuous growth rate?
- The growth rate is the coefficient of t
- Growth rate = 0.4 or 40%

- Graph the function (predict what it looks like)

Converting Between Forms

- Change to the form Q = A*Bt
- We know B = ek
- Change to the form Q = A*ek*t
- We know k = ln B (Why?)

Continuous Growth Rates

- May be a better mathematical model for some situations
- Bacteria growth
- Decrease of medicine in the bloodstream
- Population growth of a large group

Example

- A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year.
- What is the formula P(t), the population in year t?
- P(t) = 22000*e.071t

- By what percent does the population increase each year (What is the yearly growth rate)?
- Use b = ek

Assignment B

- Lesson 5.3B
- Page 417
- Exercises 65 – 85 odd

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